Applications of Derivatives
Differential Calculus-2
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Grade None
Question:
Let a function $f$ be defined as $f(x) = \begin{cases} \frac{|x-1|}{x^2+1} & \text{if } x > -1 \\ x^2 & \text{if } x \leq -1 \end{cases}$. Then the number of critical point(s) on the graph of this function is/are:
Step-by-Step Solution
Key Concept: The location of the minimum of a parabola $f(x) = ax^2 + bx + c$ is determined by the sign of $b$ when $a > 0$.
The function $f(x)$ is defined piecewise with different expressions for $x \geq 1$, $-1 0, b > 0, a > 0, b 0$, the vertex (minimum) is at $x = -\frac{b}{2a} 0$. The sign of $b$ determines whether the axis of symmetry lies to the left or right of the origin.
Correct Answer: 1